The thesis is based on the research work that was carried out to investigate Finite Element Analysis (FEA) using Artificial Neural Networks (ANN). A novel ANN model, Finite Element-based Neural Networks (FE-based NN) was proposed and applied to dynamic problems in mechanics. Firstly the variational approach to a functional in solid mechanics and the structural analogies between FEM and ANN were introduced. The computation energy functional of the FE-based NN was defined. Furthermore the architecture of FE-based NN was constructed and its algorithm was derived with the variational approach to the computational energy functional. The convergence of the FE-based NN was proved and the range of the main parameters was determined. The dynamic analysis of a beam element structure was considered as an application evaluator. The index of speedup was investigated for the measure of the computational efficiency of the FE-based NN. The simulation results were promising, which were verified by the experiment results and the computations with the commercial software package, ANSYS for the finite element analysis. Finally, the conclusion and the recommendations for further work of the investigation were discussed.